Vanishing of Ext and Tor over fiber products

Notes on Tor and Ext

1. Basic homological algebra 1 1.1. Chain complexes 2 1.2. Maps and homotopies of maps of chain complexes 2 1.3. Tensor products of chain complexes 3 1.4. Short and long exact sequences 3 1.5. Dual cochain complexes and Hom complexes 4 1.6. Relations between ⊗ and Hom 4 2. The universal coefficient and Künneth theorems 5 2.1. Universal coefficients in homology 5 2.2. The Künneth theorem 6 2.3. ...

Ext, Tor and the Uct

Symmetry of Ext Vanishing over Non-commutative Rings

It is proved for certain graded non-commutative rings that if M and N are graded bi-modules, finitely generated from either side, then the Ext groups between M and N vanish from some step if and only if the Ext groups between N and M vanish from some step. Among the rings in question are twisted homogeneous coordinate rings of elliptic curves.

On a non-vanishing Ext

The existence of valuation domains admitting non-standard uniserial modules for which certain Exts do not vanish was proved in [1] under Jensen’s Diamond Principle. In this note, the same is verified using the ZFC axioms alone. In the construction of large indecomposable divisible modules over certain valuation domainsR, the first author used the property thatR satisfied Ext1R(Q,U) 6= 0, where ...

Symmetry Theorems for Ext Vanishing

It was proved by Avramov and Buchweitz that if A is a commutative local complete intersection ring with finitely generated modules M and N , then the Ext groups between M and N vanish from some step if and only if the Ext groups between N and M vanish from some step. This paper shows that the same is true under the weaker conditions that A is Gorenstein and that M and N have finite complete int...